Be sure to read “TIPS FOR COMPLETING THE PROJECTS” for instructions on how your project should be done.
1) A steel pipe is carried down a hallway 9 feet wide. At the end of the hall, there is a right angled turn into a narrower hallway 6 feet wide.
a) Determine an expression for the space available in the turn in which to carry the pipe. This space is represented by the diagonal line in the picture below. The available space will change based on the value of \(\theta\). Your expression should be in terms of \(\theta\).
b) Use GRAPH (or equivalent) graphing software to show the graph of the expression you found in Part (a) for \(0<\theta<\frac{\pi}{2}\).
c) Use your graphing calculator to find the minimum value on the graph. What are the \(x\) and \(y\) coordinates of this point? What does this represent in the context of the original problem?
d) Explain why the minimum value of \(y\) determines the maximum length of a pipe that may be carried around the corner.
(10 points)
2) A rain gutter is to be constructed by folding up the edges of a metal sheet. The metal sheet will be divided into thirds, so that one third is folded up on each side and one third is left in the middle. The width of the metal sheet is initially 30 cm., so that each third will be 10 cm. wide.
a) Find an expression for the cross-sectional area of the gutter in terms of \(\theta\).
b) Use GRAPH (or equivalent) graphing software to show the graph of the expression you found in Part (a) for \(0<\theta<\pi\).
c) Use your graphing calculator to find the maximum value on the graph. What are the \(x\) and \(y\) coordinates of this point? What does this represent in the context of the original problem?
Express the value of \(\theta\) in both degrees and radians that will maximize the area of the trapezoid in the figure.
(10 points)